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The divergence theorem is a theorem in vector calculus which relates the surface integral to the divergence inside the surface. Mathematically, it is stated as

\oiint$ {\scriptstyle S } $ $ \vec{F} \cdot \ \mathrm{d} \vec{s} = \iiint_D \nabla \cdot \vec{F} \,\mathrm{d}V $

where D is the volume of the region enclosed by the surface and S is the projection of the surface onto the plane.

The divergence theorem holds in other dimensions as well; for example, for a flux integral in two dimensions, the divergence theorem becomes

$ \oint_S \vec{F} \cdot \vec{n} \ \mathrm{d} s = \iint_D \nabla \cdot \vec{F} \ \mathrm{d} A $

which is a version of Green's theorem.

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